Emmy Noether

"In the realm of algebra, in which the most gifted mathematicians have
been busy for centuries, she discovered methods which have proved
of enormous importance... Pure mathematics is, in its way, the
poetry of logical ideas. ... In this effort toward logical beauty, spiritual
formulas are discovered necessary for deeper penetration into
the laws of nature."
--- Albert Einstein, in a tribute to Emmy Noether
[NYT1935ae]

Noether's work is of paramount importance to
physics and the interpretation of fundamental laws in terms of group theory.
--- Feza Gursey [encp1983nj]

Proved that a physical system described by a Lagrangian
invariant with respect to the symmetry transformations of a Lie group has,
in the case of a group with a
finite (or countably infinite) number of independent, infinitesimal generators,
a conservation law
for each such generator, and certain `dependencies' in the case
of a larger infinite number of generators.
The latter case applies, for example, to
the general theory of relativity and gives the Bianchi identities.
These `dependencies'
lead to understanding of energy-momentum conservation in the general theory.
Her paper
proves both the theorems described above and
their converses.
These are collectively referred to by physicists as
Noether's Theorem.

The key to the relation of symmetry laws to conservation laws is Emmy Noether's celebrated Theorem. ... Before
Noether's Theorem the principle of conservation
of energy was shrouded in mystery, leading to the obscure physical systems of
Mach and Ostwald. Noether's simple and profound mathematical formulation did
much to demystify physics.
--- Feza Gursey [encp1983nj]

The main body of her work was in the creation of modern abstract
algebra. As the topologist P. S. Alexandrov wrote

It was she who taught us to think in terms of simple and general
algebraic concepts - homomorphic mappings, groups and rings with operators,
ideals ...theorems such as the `homomorphism and isomorphism theorems',
concepts such as the ascending and descending chain conditions for subgroups and ideals,
or the notion of groups with operators were first introduced by Emmy Noether and have
entered into the daily practice of a wide range of mathematical disciplines.
... glance at Pontryagin's work on ..continuous groups, Kolmogorov on
... combinatorial topology ..., ... Hopf on continuous mappings, ...
van der Waerden on algebraic geometry, ... to sense the influence of
Emmy Noether's ideas. This influence is also keenly felt in H. Weyl's
book Gruppentheories und Quantenmechanik. ---[en1981ad]

Specifically, Nathan Jacobson writes

Abstract algebra can be dated from the publication of
two papers by Noether, the first a joint paper
with Schmeidler and .. a truly monumental work
Idealtheorie in Ringbereichen [which] belongs to one of the
mainstreams of abstract algebra, commutative ring theory, and
may be regarded as the first paper in this vast subject ...
[encp1983nj]

The theory of non-commutative algebras and their representations was
built up by Emmy Noether in a new unified, purely conceptual manner
by making use of all the results that has been accumulated by the
ingenious labors of decades by Frobenius, Dickson, Wedderburn and
others. ---[sm1935hw]

She was never elected to the Königl. Gesellschaft
der Wissenschaften zu Göttingen
.[h1970cr]
Her great 1918 paper on symmetries and conservation
laws was communicated to the Gesellschaft by Felix Klein.

Auguste Dick raises interesting questions regarding the fact that Noether was
never appointed to a paid position in the faculty of the University of
Göttingen:

"How was it then that in her academic career she did not go beyond the [unpaid] level of nicht-beamteter ausserordentlicher
Professor? ... Was it because she was Jewish? There were several Jewish
Ordinarii in Göttingen. Was it because she was a member of the social-
democratic party? ... Or was it her firm stance as a pacifist that was frowned
upon? ..." -- [en1981ad]

As a Jewish woman, in 1933 Emmy Noether was fired from her position as a privat docent in
Göttingen. By decree no Jew was allowed to teach
after Hitler came to power. (In 1934
women were dismissed from University
posts.)

Hermann Weyl wrote about her in this period

" A stormy time of struggle like this one we spent in Göttingen in the
summer of 1933 draws people closely together; thus I have a vivid recollection
of these months. Emmy Noether - her courage, her frankness, her unconcern about
her own fate, her conciliatory spirit - was in the midst of all the hatred and
meaness, despair and sorrow surrounding us, a moral solace."
[sm1935hw]

Part of a Letter to the Editor of the New York Times that Albert Einstein
wrote on the occasion of her untimely death:

``The efforts of most human beings are consumed in the struggle for
their daily bread, but most of those who are, either through fortune
or some special gift, relieved of this struggle are largely absorbed in
further improving their worldly lot. ... There is, fortunately, a minority composed of those
who recognize early in their lives that the most beautiful and satisfying
experiences open to humankind are not derived from the outside, but are
bound up with the development of the individual's own feeling,
thinking and acting. The genuine artists, investigators and thinkers
have always been persons of this kind. However inconspicuously the life of
these individuals runs its course, none the less the fruits of their
endeavors are the most valuable contributions which one generation can
make to its successors.''[NYT1935ae]